Intersctions and Collisions{cal3}

Know:

1. Intersection (w/ @param): $r_1(t)=r_2(s)$
2. Collision btwn 2 @params $r_1(t)=r_2(t)$
3. Intersection btwn @param w/ EQ $r(t)=<x(t),y(t)>$
4. Scalar vs vector projection!

• Be comfy with changing @param to @ EQ to @vector

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Making orthognal vect in 2D?

• No cross product! (only >3 dimensions)
  • Use following formula $\vec{v}=<a,b>$ then $\vec{v}=<b,-a>$ resultant orthgonal vector
  • swap the X and Y components and negate the new Y component. So { x, y } becomes { y | -x }.

Intersctions and collisions

eg1) $L_1(t)=<3t,4t+1>,L_2(t)=<2t,3t+1$ Do both of these points intersect or collide?

• To intersect:
  • $L_2(t)=L_1(t)$ → $<3s,4s+1>=<2t,3t+1>$ Sets up sys of eq → solve for s&t → ans!
• To collision: They must be at the same time
• All collsions are intersections but !OPA

eg2) L_1(t) = <0,t>, L_2(t) = <0,t+1>, no collsions but intersetcts at all points b/c nvr meets (no solutions when solve for systems of EQ)

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• Find intersctions/collsions of some $r_1=<2\sin t,2\cos t>,r_2=t<1,1>+<2,3>$

  • Where formula for intersection: $r_1(t)=r_2(s)$
  • Collsion: $r_1(t)=r_2(t)$

• Solving! (?)

  • Get sys of eq:
    • $2sint=t+2$
    • $2cost=t+3$
  • Set 2sin(t) - 2cos(t) = -1, and sint-cost=-1/2

• Shortcut!

• $r_2(t)=<t+2,t+3>,x^2+y^2=4$

• By defining intersection: (Same thing when have only 1 @param)

  • $r(t)=x(t)=t+2,y(t)=t+3$ (sys of eq)
  • Then $x(t{)}^2+y(t{)}^2=4$, then expanding must have: $t^2+4t+4+t^2+6t+9=4$